#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\begin_preamble
\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
\let\endchangemargin=\endlist
\pagenumbering{roman}
\usepackage{color}
\definecolor{commentgreen}{RGB}{0,94,11}
\definecolor{darkblue}{rgb}{0,0,0.75}
\definecolor{darkred}{rgb}{0.6,0,0}
\end_preamble
\use_default_options true
\begin_modules
customHeadersFooters
minimalistic
todonotes
figs-within-sections
\end_modules
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype true
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command biber
\index_command default
\paperfontsize default
\spacing onehalf
\use_hyperref true
\pdf_title "Hidden Markov Models"
\pdf_author "Andy Pack"
\pdf_subject "EEEM030"
\pdf_keywords "EEEM030"
\pdf_bookmarks true
\pdf_bookmarksnumbered false
\pdf_bookmarksopen false
\pdf_bookmarksopenlevel 1
\pdf_breaklinks false
\pdf_pdfborder true
\pdf_colorlinks false
\pdf_backref false
\pdf_pdfusetitle true
\papersize default
\use_geometry true
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine biblatex
\cite_engine_type authoryear
\biblio_style plain
\biblio_options urldate=long
\biblatex_bibstyle ieee
\biblatex_citestyle ieee
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date true
\justification true
\use_refstyle 1
\use_minted 0
\index Index
\shortcut idx
\color #008000
\end_index
\leftmargin 1.8cm
\topmargin 2cm
\rightmargin 1.8cm
\bottommargin 2cm
\secnumdepth 3
\tocdepth 3
\paragraph_separation skip
\defskip medskip
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle fancy
\listings_params "language=Python,breaklines=true,frame=tb,otherkeywords={self},emph={State},emphstyle={\ttb\color{darkred}},basicstyle={\ttfamily},commentstyle={\color{commentgreen}\itshape},keywordstyle={\color{darkblue}},emphstyle={\color{red}},stringstyle={\color{red}}"
\bullet 1 0 9 -1
\bullet 2 0 24 -1
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Title
\size giant
Hidden Markov Model Analysis
\end_layout
\begin_layout Author
Andy Pack
\end_layout
\begin_layout Standard
\begin_inset VSpace 15pheight%
\end_inset
\end_layout
\begin_layout Standard
\align center
\begin_inset Graphics
filename surrey.png
lyxscale 15
width 40col%
\end_inset
\end_layout
\begin_layout Standard
\begin_inset VSpace vfill
\end_inset
\end_layout
\begin_layout Standard
\noindent
\align center
EEEM030
\begin_inset Newline newline
\end_inset
January 2021
\size large
\begin_inset Newline newline
\end_inset
Department of Electrical and Electronic Engineering
\begin_inset Newline newline
\end_inset
Faculty of Engineering and Physical Sciences
\begin_inset Newline newline
\end_inset
University of Surrey
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Abstract
abstract
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FloatList figure
\end_inset
\begin_inset CommandInset toc
LatexCommand lstlistoflistings
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Right Footer
Andy Pack / 6420013
\end_layout
\begin_layout Left Footer
January 2021
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
pagenumbering{arabic}
\end_layout
\begin_layout Plain Layout
\backslash
setcounter{page}{1}
\end_layout
\end_inset
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Standard
The Markov model is a stochastic model for analysing random processes.
The model describes a system of multiple states and the transitions between
them; a key feature is the
\emph on
Markov property
\emph default
which states that the future state of the system depends only on the current
state
\begin_inset CommandInset citation
LatexCommand cite
key "towards-data-science-markov-intro"
literal "false"
\end_inset
.
This is also described as the system being
\emph on
memoryless
\emph default
.
\end_layout
\begin_layout Standard
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
Described markov chains
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
Difference with hidden markov models
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Flex TODO Note (inline)
status open
\begin_layout Plain Layout
Application to machine learning and speech in particular
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Brief
\end_layout
\begin_layout Standard
This work analyses a continuous-density hidden Markov model of 2 emitting
states describing a 1D observation space.
Various statistics and calculations will be conducted on the model including
forward, backward, occupation and transition likelihoods.
Using these, an iteration of training will be completed using the Baum-Welch
equations resulting in a new set of transition probabilities and re-estimated
output probabilities.
\end_layout
\begin_layout Standard
The model is specified as follows; the transition probabilities are described
by entry (
\begin_inset Formula $\pi$
\end_inset
) and exit (
\begin_inset Formula $\eta$
\end_inset
) probabilities for each state along with a matrix of state-to-state transitions
(
\begin_inset Formula $a$
\end_inset
),
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
A=\left\{ \pi_{j},a_{ij},\eta_{i}\right\}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\pi_{j}=\left\{ 0.44,0.56\right\}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
a_{ij}=\left[\begin{array}{cc}
0.92 & 0.06\\
0.04 & 0.93
\end{array}\right]
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\eta_{i}=\left\{ 0.02,0.03\right\}
\]
\end_inset
\end_layout
\begin_layout Standard
These, in combination, describe a state topology than can be seen graphically
in figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:State-topology"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
As previously mentioned, the states are described 1D continuous probability
density functions (PDF) of a Gaussian profile.
Each has an associated mean and variance as seen below,
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
State,
\begin_inset Formula $i$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
Mean,
\begin_inset Formula $\mu_{i}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.00
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
4.00
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
Variance,
\begin_inset Formula $\varSigma_{i}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.44
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.49
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Standard
Finally, a set of observations from the above model are provided below,
\end_layout
\begin_layout Standard
\noindent
\align center
\begin_inset Formula
\[
\mathcal{O}=\left\{ 3.8,4.2,3.4,-0.4,1.9,3.0,1.6,1.9,5.0\right\}
\]
\end_inset
\end_layout
\begin_layout Standard
These are used as the basis for training the model using the Baum-Welch
equations.
\end_layout
\begin_layout Section
Implementation
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename StateTopology.png
lyxscale 50
width 65col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
State topology for the specified Markov Model
\begin_inset CommandInset label
LatexCommand label
name "fig:State-topology"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Section
Results
\end_layout
\begin_layout Subsection
Output Probability Density Functions
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename res/pdfs.png
width 50col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability density functions for both states across the observation space
\begin_inset CommandInset label
LatexCommand label
name "fig:PDFs"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Observation Probability Densities
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
t
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $o_{t}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $b_{i}\left(o_{t}\right)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=1$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=2$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
3.8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.022
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.55
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
4.2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0095
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.55
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
3.4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.045
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.39
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
-0.4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.17
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.5
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.25
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0063
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
3.0
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.083
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.21
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.29
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0016
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.25
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0063
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
5.0
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0013
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.21
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability densities for each observation from each state to (2 s.f)
\begin_inset CommandInset label
LatexCommand label
name "tab:obs-prob-dens"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename res/pdfs-w-obs.png
width 50col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Previous PDFs (figure
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PDFs"
plural "false"
caps "false"
noprefix "false"
\end_inset
) with observations highlighted
\begin_inset CommandInset label
LatexCommand label
name "fig:PDFs-w-obs"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Forward Procedure and Likelihoods
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
t
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\alpha_{t}\left(i\right)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=1$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=2$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $9.61\times10^{-3}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.306
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.00\times10^{-4}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.156
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.89\times10^{-4}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0573
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $4.31\times10^{-4}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $8.01\times10^{-11}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $9.95\times10^{-5}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.64\times10^{-7}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $7.59\times10^{-6}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.26\times10^{-6}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.06\times10^{-6}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.59\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $4.76\times10^{-7}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $7.99\times10^{-10}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $5.63\times10^{-10}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $6.02\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Forward likelihoods for each state at each time step
\begin_inset CommandInset label
LatexCommand label
name "tab:Forward-likelihoods"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Backward Procedure and Likelihoods
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
t
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\beta_{t}\left(i\right)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=1$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=2$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $6.58\times10^{-11}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $6.25\times10^{-10}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.93\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.23\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $6.90\times10^{-8}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $3.00\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $4.45\times10^{-7}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.11\times10^{-8}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.93\times10^{-6}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $3.02\times10^{-7}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.51\times10^{-5}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.15\times10^{-6}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $9.30\times10^{-5}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $3.77\times10^{-5}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $3.93\times10^{-4}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $5.73\times10^{-3}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.02
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.03
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Backward likelihoods for each state at each time step
\begin_inset CommandInset label
LatexCommand label
name "tab:Backward-likelihoods"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Observations Likelihood
\end_layout
\begin_layout Subsection
Occupation Likelihoods
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
t
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\gamma_{t}\left(i\right)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=1$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $i=2$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.003295
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9967
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.003055
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9969
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.1040
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.8960
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9999...
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $8.818\times10^{-9}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9997
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0002579
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9925
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.007517
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9995
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0005095
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9761
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.02386
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.05868
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.9413
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Occupation likelihoods for each state at each time step (4 s.f.)
\begin_inset CommandInset label
LatexCommand label
name "tab:Occupation-likelihoods"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Baum-Welch Re-estimations
\end_layout
\begin_layout Subsubsection
Output Parameters
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
filename res/re-est-pdfs.png
width 50col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Probability density functions following re-estimation of parameters
\begin_inset CommandInset label
LatexCommand label
name "fig:Re-estimated-PDFs"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
State Transitions
\end_layout
\begin_layout Section
Discussion
\end_layout
\begin_layout Section
Conclusion
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:bibliography"
\end_inset
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "references"
options "bibtotoc"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Section
\start_of_appendix
Intermediate Results
\end_layout
\begin_layout Section
Source Code
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../constants.py"
lstparams "caption={Constants file defining initial state transition probabilities and PDF parameters},label={constants-listing}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../maths.py"
lstparams "caption={Maths utility file with definition for a gaussian},label={maths-listing}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../markov.py"
lstparams "caption={Markov model object defining forward and backward procedure, occupation likelihoods, Baum-Welch equations},label={markov-listing}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../markovlog.py"
lstparams "caption={Child Markov model re-implementing calculations using log probability and likelihood},label={log-markov-listing}"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Standard
The development of the model behind this report was completed using Jupyter
Notebook.
The used notebook can be seen formatted in plain text below, the relevant
mark scheme elements are referenced in brackets.
\end_layout
\begin_layout Standard
\begin_inset CommandInset include
LatexCommand lstinputlisting
filename "../notebook.py"
lstparams "caption={Plain output of Jupyter Notebook used for development and debugging},label={notebook-listing}"
\end_inset
\end_layout
\end_body
\end_document